$A$ wagon of $200\, kg$ is moving on a smooth track with a velocity of $2\, m/s$. $A$ man of $80\, kg$ runs in the wagon with a velocity such that the speed of the centre of mass of the system is zero. Find the relative velocity of the man with respect to the wagon in $m/s$.

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side length $d$. The ratio $I_{0} / I_{A}$ of the moment of inertia $I_{0}$ of the system about an axis passing through the centroid and perpendicular to the plane of the triangle,to the moment of inertia $I_{A}$ about an axis passing through the center of any one of the spheres and perpendicular to the plane of the triangle,is:

$A$ heavy rod of weight $W$ is held in a horizontal position by two men at its ends. If one man suddenly lets go,what force will the other man feel?

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same $(M)$ and the mass of the threads are negligible,the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at $\frac{r}{2}$ distance from the end $A$ of the handle will be ....... $Mr^2$?

Four spheres,each of mass $M$ and diameter $2a$,are placed at the corners of a square of side $b$. The moment of inertia of this system about an axis along one of the sides of the square is:

In the given figure,a ring of mass $m$ is kept on a horizontal surface,and a body of equal mass $m$ is attached through a string wound on the ring. When the system is released,the ring rolls without slipping. Consider the following statements and choose the correct option.
$(i)$ Acceleration of the centre of mass of the ring is $\frac{g}{3}$.
$(ii)$ Acceleration of the hanging particle is $\frac{2g}{3}$.
$(iii)$ Frictional force (on the ring) acts in the forward direction.
$(iv)$ Frictional force (on the ring) acts in the backward direction.